1. Colorado Center for Astrodynamics Research The University of Colorado ASEN 5070 OD Accuracy Assessment OD Overlap Example Effects of eliminating parameters from the solution list Project Report ASEN 5070 LECTURE 34, 35, 36 11/20/09, 11/30/09, 12/2/09,

2. Colorado Center for Astrodynamics Research The University of Colorado Techniques for OD Solution Accuracy Assessment I. Examine P, the estimation error variance-covariance matrix 1.Plot tracking data residuals Are they Gaussian? What is the mean and RMS for each data type? If these agree with the apriori data statistics, one can believe that P actually represents the estimation errors – this will probably never happen. 2. In general the correlations in P will be nearly correct but the variances will be optimistic unless process noise has been added and properly calibrated (tuned). 3. Do solution overlaps and compute statistics on them Any common biases will cancel 4. If the spacecraft has laser tracking compare spacecraft slant range computed from OD solutions with withheld laser range measurements. Lasers are generally accurate to 1 or 2 cm.

3. Colorado Center for Astrodynamics Research The University of Colorado Techniques for OD Solution Accuracy Assessment 5. Map P to the high elevation period and compare range standard deviation with laser range residuals computed in part 4. Use this difference as a scaling factor for P, i.e., if is half as large as RMS of computed range differences, increase P by a factor of 4. 6. If the spacecraft carries an altimeter, examine cross over differences over the ocean or any point where the altimeter measures accurately and local surface elevation is constant. surface ellipsoid geoid Orbit #1 Orbit #2 h s g R r1r1 h = altimeter measurement r 1 -h 1 = R+g+s+e 1 = C 1 r 2 -h 2 = R+g+s+e 2 = C 2 r 1, r 2 = OD solutions, radius from geocenter e 1, e 2 =orbit error C 1 -C 2 = orbit error less common biases r2r2 x (geocenter)

4. 4 Copyright 2006 In this study, an overlap comparison of solutions for the term project was performed for tracking data obtained from two different overlapping sets of range and range rate data. This data was divided into two arcs as shown in Figure A. The batch filter was used to generate a best estimate for the state and the covariance at the time corresponding to the two data set epochs. The covariance was mapped to the desired time using the equation Figure A Two arcs used in the overlap comparison Term Project overlap study

5. 5 Copyright 2006 Figure 1 Overlap comparison using Range and Range Rate A comparison was desired during the overlap time period between 1.5 and 3.5 hours. Plots were generated with one standard deviation (1σ) magnitudes from P k, and the differences in the solution overlaps for this period. They are given in the RIC (radial, in- track, cross-track) coordinate system in Figure 1. The actual differences in the radial and in-track directions generally fell within the one  values obtained from the early and the late cases. Term Project overlap study

6. 6 Copyright 2006 The batch algorithm was also used with each data type separately. Figure 2 Overlap comparison using Range Figure 3 Overlap comparison using Range Rate Term Project overlap study

7. 7 Copyright 2006 Note that in all cases the cross-track standard deviation and overlap differences exhibit a once per rev sinusoidal behavior. In this case the maximum differences occur when the satellite is at its highest latitude, and the minimum is at the equatorial crossing. To first order, only a difference in inclination or right ascension of the ascending node will cause a cross-track deviation and these will be 90  out of phase. A difference in node will cause a maximum deviation at the equatorial crossing while a difference in inclination will have a maximum effect on cross track deviation at maximum latitude. Therefore, it is concluded that this difference is primarily caused by differences in the inclination of the two orbits. Term Project overlap study

8. 8 Copyright 2006 In general the formal or data noise covariance matrix would not be this indicative of actual errors in the orbit determination solution because errors would arise from many sources which are unmodeled or inaccurately modeled. In this case the mathematical model contains all quantities affecting the satellite’s motion and there are no measurement errors except random noise. Consequently, the formal covariance matrix should be an accurate measure of estimation error. Term Project overlap study

9. 9 Copyright 2006 Term Project overlap study

10. 10 Copyright 2006 Term Project overlap study

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12. 12 Copyright 2006 Term Project overlap study

13. 13 Copyright 2006 Term Project overlap study

14. 14 Copyright 2006 Term Project overlap study

15. 15 Copyright 2006 Summary Overlap studies are a consistency but not an absolute accuracy check These results demonstrate that in general range is a stronger data type than range rate. This can be explained by noting that range measurements, which for this study are accurate to 0.01 m at 20 sec intervals, essentially provides the information equivalent to range rate with an accuracy of 0.01/20 or 0.5 x 10 -3 m/sec. Hence, range observations provide not only a measure of the range, but also a measure of range rate. Each range rate measurement is accompanied by an unknown value of range. This situation could be mitigated if the range rate were generated by continuous count Doppler. In this case it could be processing as accumulated or integrated range. This would result in only one unknown epoch value of range and would increase the strength of the range rate data. It is useful to examine covariances in alternate coordinate systems Term Project overlap study

16. 16 Copyright 2006 Effects of eliminating parameters from the solution list of the term project What will be the effect of fixing J2 at ½ its value and not solving for it? How will this affect 1.The residuals? 2.The estimation error covariance matrix variations for position and velocity? 3.Variances for the remaining parameters in the solution list? 4.Correlation coefficients between J2 and the position and velocity? 5.Correlation coefficients between J2 and the remaining solved for parameters?

17. 17 Copyright 2006 Post-fit residuals when estimating J2 as in the term Project

18. 18 Copyright 2006 Post-fit residuals when fixing J2 at ½ its actual value

19. 19 Copyright 2006 Effects of eliminating parameters from the solution list of the term project, cont. Trace of Position variances (P(1:3,1:3)): J 2 estimated: 0.000416001592394197 m 2 J 2 fixed: 0.000340165200752961 m 2 Trace of Velocity variances (P(4:6,4:6)): J 2 estimated: 3.87987125981823e-10 m 2 /s 2 J 2 fixed: 3.33395326676567e-10 m 2 /s 2

20. 20 Copyright 2006 Effects of eliminating parameters from the solution list of the term project, J 2 estimated Columns 1 through 3 5.66248951105043e-05 -6.34995290877777e-05 5.83676474867691e-05 -0.715470305515882 0.000139107444083783 -0.000147700652606973 0.522626641263219 -0.843782063029082 0.000220269253199909 -0.807236306276535 0.893453128315322 -0.750828480428999 0.680645690588889 -0.96548285467716 0.936698223473284 0.392935147605768 -0.819102417018464 0.971368676263619 0.716300177506644 -0.422901518948097 0.351004925603219 Columns 4 through 6 -5.24764107312374e-08 7.39743457168762e-08 3.0263270986426e-08 9.10346423198167e-08 -1.64465980897838e-07 -9.88791391360134e-08 -9.62669934551078e-08 2.0078578570408e-07 1.47554452172925e-07 7.46310345033749e-11 -1.07315035843011e-10 -6.05179116137273e-11 -0.860091223256874 2.08599208551714e-10 1.32038618166525e-10 -0.684436116140288 0.893209551987995 1.04756882926734e-10 -0.329099080725397 0.461997852034025 0.239062666054139 Column 7 1.3181716058567e-12 -1.21979716308379e-12 1.27398213805882e-12 -6.95279762260685e-16 1.63180955939867e-15 5.983788442396e-16 5.98062231305506e-20 Position, velocity, and J 2 Variance-covariance, and correlation matrix. Upper triangular portion – covariances Diagonals – variances Lower triangular portion – correlation coefficients

21. 21 Copyright 2006 Effects of eliminating parameters from the solution list of the term project, J 2 fixed Columns 1 through 3 2.82921847411582e-05 -3.76090896815745e-05 3.15618952055929e-05 -0.658339808283788 0.000115350352103804 -0.000123860986228135 0.423275868659951 -0.822656910466205 0.000196522663907999 -0.869076488582465 0.883993092809911 -0.7243857756462 0.571507757196623 -0.958652500072026 0.934340316962549 0.331168664496907 -0.815921027632663 0.975734846624612 4.17715956478505e-06 -1.86727466352177e-06 1.50986127545734e-06 Columns 4 through 6 -3.79783381401357e-08 3.93284681315099e-08 1.74839479237644e-08 7.80015166618908e-08 -1.33205553046635e-07 -8.69790405985169e-08 -8.34297511663449e-08 1.69458387259265e-07 1.35767293813083e-07 6.74977702869742e-11 -9.01904881666218e-11 -5.40668944167407e-11 -0.848524459471428 1.67379905007772e-10 1.16672317146435e-10 -0.663024667331318 0.90857156259011 9.85176513818207e-11 -1.37879730334042e-06 2.0935171284686e-06 9.62544905311382e-07 Column 7 2.22184781121686e-23 -2.0054769127344e-23 2.11662230204251e-23 -1.13277886800604e-26 2.7084958657571e-26 9.55384133773851e-27 9.99999999982403e-31 Position, velocity, and J 2 Variance-covariance, and correlation matrix. Upper triangular portion – covariances Diagonals – variances Lower triangular portion – correlation coefficients

22. 22 Copyright 2006 Effects of eliminating parameters from the solution list of the term project Conclusions 1. Fixing J 2 at an incorrect value increases the tracking residuals and introduces systemic signatures 2. Fixing J 2 eliminates the correlations between J2 and all other estimated parameters 3. Fixing J 2 at any value reduces the variances on all estimated parameters that are correlated with J 2. The filter has fewer parameters to solve for so it thinks it can solve for these parameters to greater accuracy 4. These statements are true for any parameter in the solution list By fixing J 2 we mean removing it from the estimation list or setting it’s a priori variance nearly to zero

23. 23 Copyright 2006 Requirements/Suggestions for Term Project Report **1. General description of the OD problem and the batch and sequential algorithms **2. Discussion of the results - contrasting the batch processor, and sequential filter. Discuss the relative advantages, shortcomings, applications, etc. of the algorithms. **3. Show plots of residuals for all algorithms. Plot the trace of the covariance for position and velocity for the sequential filter for the first iteration. You may want to use a log scale. **4. When plotting the trace of P for the position and velocity, do any numerical problems show up? If so discuss briefly how they may be avoided. **5. Contrast the relative strengths of the range and range rate data. Generate solutions with both data types alone for the batch and discuss the solutions. How do the final covariances differ? You could plot the two error ellipsoids for position. What does this tell you about the solutions and the relative data strength? **6. Why did you fix one of the stations? Would the same result be obtained by not solving for one of the stations i.e., leaving it out of the solution list? Does it matter which station is fixed? **7. A discussion of what you learned from the term project and suggestions for improving it. **Required items for the final report

24. 24 Copyright 2006 Requirements/Suggestions for Term Project Report Extra Credit Items 8. Code the Extended Kalman Filter 9. How does varying the a priori covariance and data noise covariance affect the solution? What would happen if we used an a priori more compatible with the actual errors in the initial conditions, i. e. a few meters in position etc. 10. Do an overlap study (see lecture 34). 11. Code the Potter algorithm and compare results to the conventional Kalman filter. 12. Solve for the state deviation vector using the Givens square root free algorithm. Compare solution and RMS residuals for range and range rate from Givens solution with results from conventional Kalman and Potter filters. 13. Plot pre and post fit residuals for the Kalman filter. Include the 1 sigma pre-fit standard deviations on the plot (See Eqn. 4.7.34 of text). 14. Convert the estimation error covariance matrix into classical orbit element and nonsingular orbit element space. What elements are most in error? Can you see a reason for this. The coordinate transformation code is being mailed to the class. 15. Examine the effects of fixing various parameters such as J2 and seeing the results on the solution and residuals. They will no longer be Gaussian.